Track your object in the Movie Clip Editor
Select the tracker and in the solve section select Link empty to track.
Switch to the the regular 3d view. Through the camera biew you'll see an empty that behaves like the tracker.
(I added the video image as background for clarity, but is not necessary)
Select the empty and snap the cursor to the empty (Ctrl+S and select Cursor to selected)
Add a Curve > Bezier object.
I edit mode (Tab) select the end control point of the curve, and snap it to the cursor, that will move the end of the curve to the empty.
Exit edit mode.
Select the Empty and while pressing shift select the curve. And enter edit mode (this is done so that the Empty becomes the target object for the next operation)
Press the space bar, type "hook" and select "Hook to Selected object".
That will make the end of the curve follow the empty.
Go to the first frame of the video (or where the movement begins), and snap the cursor to the empty (Shift S > snap cursor to selected). Don't worry if the other segment seems to unhook from the empty momentarily
Then select the curve, enter edit mode, select the first segment of the curve and snap it to the empty (snap selection to cursor). (don't worry if the other segment seems to unhook from the empty momentraily)
Note that the curve will track the beginning and end points but not necessarily curve along the way... you might need to adjust that manually.